The tensor curvature \(B_{\alpha\beta}\) and the Riemann-Christoffel tensor \(R_{\cdot\delta\alpha\beta}^{\gamma}\) are two manifestations of the curvature of a surface, but arise in different ways. The curvature tensor \(B_{\alpha \beta}\) arises from the fact that the surface covariant basis \(\mathbf{S} _{\alpha}\) undulates with the surface and, as a result, its derivatives \(\nabla_{\alpha}\mathbf{S}_{\beta}\) have normal components that form the curvature tensor \(B_{\alpha\beta}\). Thus, the curvature tensor \(B_{\alpha \beta}\), which does not vanish for any surface other than a plane, is a characteristic of the manner in which the surface is embedded in the ambient space.

The Riemann-Christoffel tensor \(R_{\cdot\delta\alpha\beta }^{\gamma}\), on the other hand, arises from within. Namely, it is a manifestation of the fact that surface covariant derivatives do not commute, i.e. More specifically, the commutator \(\nabla_{\alpha}\nabla_{\beta}-\nabla _{\beta}\nabla_{\alpha}\) applied to a variant \(T^{\gamma}\) with a surface superscript is governed by the rule where the Riemann-Christoffel tensor \(R_{\cdot\delta \alpha\beta}^{\gamma}\) is defined by the equation The Riemann-Christoffel tensor \(R_{\gamma\delta\alpha\beta}\) with the lowered first index is given by Demonstration of the above identities is left as an exercise.

We must again draw your attention to the crucial intrinsic property of the Riemann-Christoffel tensor. Since the Christoffel symbol can be expressed strictly in terms of the metric tensor and its derivatives, i.e. the same is true for the Riemann-Christoffel tensor. Meanwhile, as we showed in Section TBD of Introduction to Tensor Calculus, the metric tensor can be calculated by measuring distances of curves within the surface. In other words, the metric tensor can (theoretically) be calculated by intrinsic means without any reference to the precise manner in which the surface is embedded in the ambient space. And, since the Riemann-Christoffel symbol can be expressed in terms of the metric tensor and its derivative, it too can be calculated by intrinsic means. Therefore, the Riemann-Christoffel tensor can carry only partial information about the curvature of the surface. Indeed, any deformation of the surface that preserves in-surface distances between points also preserves the Riemann-Christoffel tensor. For example, a flag made of inextensible material is characterized by a vanishing Riemann-Christoffel tensor regardless of its shape.

Meanwhile, the flag is decidedly not flat and, correspondingly, the curvature tensor \(B_{\alpha\beta}\) does not vanish. Thus, clearly, the curvature tensor \(B_{\alpha\beta}\) carries at least some information not contained in the Riemann-Christoffel tensor. But what about the inverse: is there any information in the Riemann-Christoffel \(R_{\cdot\delta\alpha\beta}^{\gamma}\) tensor not contained in the curvature tensor? The answer, provided by the Gauss equations, is no. As we have already stated on a number of occasions, the Gauss equations read and therefore tells us that the Riemann-Christoffel tensor can be constructed from the information contained in the curvature tensor.

From the equation it is easily seen that the Riemann-Christoffel tensor \(R_{\gamma\delta \alpha\beta}\) is antisymmetric in \(\alpha\) and \(\beta\), i.e. Far less obvious is the fact that the Riemann-Christoffel tensor is symmetric with respect to switching the first two indices with the last two, i.e. The proof of this relationship is left as an exercise. From the above symmetries, it is easy to show that the Riemann-Christoffel tensor \(R_{\gamma\delta\alpha\beta}\) is also anti-symmetric in \(\gamma\) and \(\delta\), i.e.

In addition to the above symmetries, the Riemann-Christoffel tensor satisfies the identity known as the first Bianchi identity. It is left as an exercise to show that, owing to the above symmetries, the Riemann-Christoffel tensor has degrees of freedom.

Meanwhile, the second Bianchi identity involves covariant derivatives of the Riemann-Christoffel tensor and reads These identities are named after the Italian mathematician Luigi Bianchi who was a colleague of Gregorio Ricci and Tullio Levi-Civita.

The Riemann-Christoffel tensor originally arose in the expression for i.e. As we mentioned before, this formula was first given by Gregorio Ricci and Tullio Levi-Civita in their 1901 paper titled M\'{ethodes de calcul differ'{e}ntiel absolu et leurs applications}. In fact, in his classic Foundations of the Theory of Surfaces in Tensor Terms, Veniamin Kagan refers to the above relationship as the Ricci identity.

A natural question is: what is the corresponding expression for and, indeed, for variants with arbitrary indicial signatures? Answering this question is the goal of this Section.

Let us begin with a variant \(T_{\gamma}\) with a single covariant index. By lowering \(\gamma\) on both sides of the identity we obtain Note that this step relied on the fact that surface indices can be juggled across the surface covariant derivative. In order for the variant \(T^{\delta}\) to appear in its matching covariant form on the right, exchange the flavors of the index \(\delta\), i.e. While this identity represents the answer to the question that we have posed, we would like, for consistency's sake, for the Riemann-Christoffel index to appear with a contravariant first index and the covariant second index. Since the Riemann-Christoffel symbol is anti-symmetric in its first two indices, i.e. we arrive at the final result

Note the pleasing structural parallel between the terms borne by the commutator \(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta} \nabla_{\alpha}\) and the terms that appear in the definition of the covariant derivative for the variants \(T^{\gamma}\) and \(T_{\gamma}\).

The next logical task is to extend the equations to second- and higher-order variants. It turns out that the result of applying the commutator \(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha}\) to a higher-order variant is the sum of characteristic terms for each index. Specifically, for a variant \(T_{\delta}^{\gamma}\) with a representative collection of surface indices, we have As always, this equation is to be understood in the sense of prescribing the appropriate term for each type of index. The proof of this equation is left as another labor-intensive, but worthwhile exercise.

Next, let us extend the commutator \(\nabla_{\alpha}\nabla_{\beta} -\nabla_{\beta}\nabla_{\alpha}\) to variants with ambient indices. It turns out that for surfaces embedded in a Euclidean space, the surface covariant derivatives commute for variants with ambient indices, i.e. For first-order variants with scalar elements, such as \(T^{i}\), this can be demonstrated inductively by considering the vector quantity As we established earlier, surface covariant derivatives commute for variants of order zero, i.e. Since the surface covariant derivative is metrinilic with respect to the ambient basis \(\mathbf{Z}_{i}\), we have which implies the equation A more general approach to demonstrate the fact that surface covariant derivatives commute for variants with ambient indices is to derive the relationship between the surface Riemann-Christoffel tensor \(R_{\gamma \delta\alpha\beta}\) and its ambient counterpart \(R_{ijkl}\). Working out this approach is left as an exercise.

Putting it all together, we can establish the result of applying the commutator \(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha}\) to variants with arbitrary indicial signature. For a tensor \(T_{j\delta} ^{i\gamma}\) with a representative collection of indices, that equation reads and, as always, it is understood as a recipe for constructing the appropriate combinations for arbitrary collections of indices.

We have finally arrived at the Gauss equations which we will now discuss from a number angles. It is captured by the equation Note, importantly, that, although the Gauss equations are most famously applied to two-dimensional surfaces, they are valid in any number of dimensions. In this Section, we will discuss on the general \(n\)-dimensional case while in the next Section, we will focus on the special case of two-dimensional surfaces.

In Exercise 2.23, this equation was derived by applying the commutator \(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha}\) to the surface covariant basis \(\mathbf{S}_{\gamma}\). However, such a derivation is applicable only to surfaces embedded in Euclidean spaces. Meanwhile, we would like to give a derivation that can be readily adapted to hypersurfaces embedded in \(n\)-dimensional Riemannian ambient spaces. To this end, instead of applying the commutator \(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta} \nabla_{\alpha}\) to the surface covariant basis, we will apply it to the shift tensor \(Z_{\gamma}^{i}\).

According to the commutator equation we have In order to establish the expression for \(\nabla_{\alpha}\nabla_{\beta }Z_{\gamma}^{i}\), recall that Thus, By the product rule, we find Now recall Weingarten's equation for the covariant derivative of the normal \(N^{i}\) With the help of Weingarten's equations, we find Switching the indices \(\alpha\) and \(\beta\) yields the expression for \(\nabla_{\beta}\nabla_{\alpha}Z_{\gamma}^{i}\) Thus, which, in combination with the equation yields the identity known as the Gauss-Codazzi equations.

The left side of the Gauss-Codazzi equations clearly consists of tangential and normal parts. In order to extract the normal part, contract both sides of the equation with the normal \(N_{i}\), which yields the equation It is known as the Codazzi equation and is usually written in the form which tells us that \(\nabla_{\alpha}B_{\beta\gamma}\) is fully symmetric in all of its subscripts.

In order to extract the tangential part of the Gauss-Codazzi equations contract both sides with the shift tensor \(Z_{i}^{\varepsilon}\) and subsequently rename \(\varepsilon\) into \(\delta\). The result is which is, indeed, the Gauss equations which we have been billing up ever since we introduced the curvature tensor.

Lowering the superscript \(\delta\) yields the covariant form which we will prefer to write with the following combination of indices

Raising the indices \(\alpha\) and \(\beta\) yields the form with a balanced number of superscripts and subscripts which is conducive to producing invariant relationships. Indeed, contracting \(\alpha\) with \(\gamma\) and \(\beta\) with \(\delta\), we find The invariant is known as the scalar curvature. If the eigenvalues of \(B_{\beta }^{\alpha}\) are denoted by \(\lambda_{1},\cdots,\lambda_{n}\), then while Therefore, in terms of the eigenvalues of \(B_{\beta}^{\alpha}\), the scalar curvature \(R\) is given by

This completes our discussion of the general \(n\)-dimensional case and we will now turn our attention to two-dimensional surfaces.

On a two-dimensional surface, the Riemann-Christoffel tensor has \(2^{4}=16\) elements. However, owing to the available symmetries there can be only four nonzero elements, i.e. Furthermore, the same symmetries dictate that these elements are related by which reduces the actual number of degrees of freedom to \(1\). Allowing the element \(R_{1212}\) to present the sole degree of freedom, we can capture the two-dimensional Riemann-Christoffel tensor with the help of the permutation symbols \(e_{\alpha\beta}\) and \(e_{\gamma\delta}\), i.e. In order to tensorize the above identity, switch from permutation systems \(e_{\alpha\beta}\) and \(e_{\gamma\delta}\) to the Levi-Civita symbols \(\varepsilon_{\alpha\beta}\) and \(\varepsilon_{\gamma\delta}\). Since where \(S\) is, of course, the determinant of the metric tensor \(S_{\alpha\beta }\), we have

By the quotient theorem, the quantity is an invariant. It is known as the Gaussian curvature and is denoted by the symbol \(K\), i.e. In terms of the Gaussian curvature, the Riemann-Christoffel tensor is given by

There are two explicit expressions for \(K\). The first one is obtained by raising the indices \(\alpha\) and \(\beta\), i.e. and subsequently contracting with \(\gamma\) and \(\delta\), which yields. or Alternatively, contract both sides of the identity with \(\varepsilon^{\alpha\beta}\varepsilon^{\gamma\delta}\), which yields or

The Gauss equations can be written in a number of special forms for two-dimensional surfaces.

Recall from Chapter TBD of Introduction to Tensor Calculus, that for any second-order system \(A_{\alpha\beta}\) in two dimensions, we have where \(A\) is the determinant of the mixed system \(A_{\cdot\beta}^{\alpha}\). Thus, for the curvature tensor \(B_{\alpha\beta}\), we have where \(B\) is the determinant of \(B_{\beta}^{\alpha}\). Meanwhile, by the Gauss equation, the combination on the left equals the Riemann-Christoffel tensor \(R_{\alpha\beta\gamma\delta}\). Thus, its alternative formulation on a two-dimensional surface is

At the same time, recall that (for reasons having nothing to do with the Gauss equations), the Riemann-Christoffel tensor \(R_{\alpha\beta\gamma\delta}\) is given by Thus, and, indeed, In words, the Gaussian curvature \(K\) equals the determinant \(B\) of the mixed curvature tensor \(B_{\beta}^{\alpha}\). This is clearly the most concise form of the Gauss equations for two-dimensional surfaces. It follows that the Gaussian curvature equals the product of the principal curvatures \(\kappa_{1}\) and \(\kappa_{2}\), i.e.

Yet another form of the Gauss equations ties together the traces of the first, second, and third fundamental tensors. Write the Gauss equations for a two-dimensional surface in the form Raising the index \(\alpha\), i.e. and subsequently contracting it with \(\delta\), we find Since \(\varepsilon_{\cdot\beta}^{\alpha}\varepsilon_{\alpha\delta} =S_{\beta\delta}\), we have After reshuffling the names of the indices and taking advantage of the symmetry of the curvature tensor, this equation can be written in the form

Recall that the metric tensor \(S_{\alpha\beta}\), the curvature tensor \(B_{\alpha\beta}\), and the tensor are sometimes referred to as the first, second, and third groundforms of the surface. Thus, the equation may be described as relating the three fundamental groundforms of the surface.

Upon raising the index \(\beta\) and contracting with \(\alpha\), as well as renaming \(\gamma\rightarrow\beta\), we arrive at the invariant equation This equation relates three invariants: the mean curvature \(B_{\alpha }^{\alpha}\), the Gaussian curvature \(K\), and the trace \(B_{\beta}^{\alpha }B_{\alpha}^{\beta}\) of the third groundform.

Recall that for a sphere of radius \(R\), the curvature tensor \(B_{\beta }^{\alpha}\) is given by Therefore,

Recall that for a cylinder of sphere \(R\), the curvature tensor \(B_{\beta }^{\alpha}\) is given by Thus, which is consistent with the fact that a cylinder can be isometrically deformed into a section of the plane.

Recall that for a torus with radii \(R\) and \(r\), the curvature tensor \(B_{\beta}^{\alpha}\) is given by Therefore,

Recall that for a surface of revolution given by the functions \(G\left( \gamma\right) \) and \(H\left( \gamma\right) \), the curvature tensor \(B_{\beta}^{\alpha}\) is given by Therefore, For the common choice \(H\left( \gamma\right) =\gamma\), we have

Finally, we take a moment to preview an important integration theorem that we will discuss in the future. The Gauss-Bonnet theorem is a fundamental result in differential geometry and topology. It states that for a closed surface, the total curvature, defined as the integral of the Gaussian curvature \(K\), depends on the genus of the surface and not its shape. The genus \(g\) of a surface is the number of topological holes. For example, the genus of a sphere is zero and the genus of a torus is one. According to the Gauss-Bonnet theorem, the total curvature is \(4\pi\left( 1-g\right) \): In particular, the total curvature is \(4\pi\) for any surface of genus zero and \(0\) for any surface of genus one.

Exercise 7.1Show that where, naturally

Exercise 7.2Derive the anti-symmetric property of the Riemann-Christoffel tensor from the identity

Exercise 7.3Demonstrate the symmetric property This can be accomplished by the approach outlined in the exercises of Chapter TBD of Introduction to Tensor Calculus. Alternatively, it can be accomplished more easily with the help of the Gauss equations. However, this approach is less general since the Gauss equations are valid only for surfaces embedded in a Euclidean space.

Exercise 7.4Demonstrate the first and the second Bianchi identities and These tasks can also be accomplished either by a direct calculation or, more easily but with less generality, with the help of the Gauss and Codazzi equations.

Exercise 7.5Show that and, similarly,

Exercise 7.6Show that, owing to the inherent symmetries, including the first Bianchi identity, the Riemann-Christoffel tensor has degrees of freedom.

Exercise 7.7Demonstrate the equation by an inductive argument. Hint: apply the commutator \(\nabla_{\alpha} \nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha}\) to the variant where \(S^{\delta}\) is an arbitrary first-order variant.

Exercise 7.8Use an inductive approach to show that surface covariant derivatives commute for variants with ambient indices, i.e. for a variant \(T_{j}^{i}\) with a representative collection of ambient indices, this statement reads

Exercise 7.9Show that for a surface embedded in a Riemannian space, the surface and ambient Riemann-Christoffel tensors \(R_{\alpha\beta\gamma\delta}\) and \(R_{ijkl}\) are related by the identity which represents a generalization of the Gauss equations, but also makes us see the latter simply as a relationship between the surface and ambient Riemann-Christoffel tensors.

Exercise 7.10Show that the equation can be written in the form which is the form in which it usually appears in sources that do not empoy index juggling.

Exercise 7.11Show that the Gaussian curvature \(K\) is given by the following equation

Exercise 7.12Demonstrate the Bieberbach formulas and where \(S\) is the determinant of \(S_{\alpha\beta}\). These formulas appeared in Ludwig Bieberbach's Differentialgeometrie published in 1932. On a historical note, Bieberbach was an ardent adherent of the Nazi cause and actively pursued the dismissal of his Jewish colleagues. This fact is gracefully overlooked by Veniamin Kagan in his Foundations of the Theory of Surfaces in Tensor Terms where he describes the above formulas as most simple and elegant.

Exercise 7.13Show that the Gaussian curvature vanishes for a cone, which is consistent with the fact that a cone can be isometrically deformed into a section of a plane.

Exercise 7.14Show that the Gaussian curvature vanishes for a tangent developable discussed in Section 6.7.1.

Exercise 7.15Consider a curve given as the graph of a function \(z=F\left( x,y\right) \) in Cartesian coordinates \(x,y,z\). Derive all of the relevant geometric objects. In particular, show that Consequently, and

The next set of exercises has to do with the following definitions. The Ricci tensor \(R_{\alpha\beta}\) is defined by Its trace is known as the scalar curvature. Finally, the Einstein tensor \(G_{\alpha\beta}\) is defined by

begin{exercise} Show that the scalar curvature is given by end{exercise}

Exercise 7.16Show that the Ricci tensor \(R_{\alpha\beta}\) is symmetric, i.e.

Exercise 7.17Show that the Einstein tensor \(G_{\alpha\beta}\) is symmetric, i.e.

Exercise 7.18Show that for a two-dimensional surface the trace of the Einstein tensor vanishes, i.e.

Exercise 7.19Show that for a two-dimensional surface

Exercise 7.20Verify the Gauss-Bonnet theorem for a sphere, i.e.

Exercise 7.21Verify the Gauss-Bonnet theorem for a torus, i.e.